Biostat Didactic Seminar Series Analyzing Binary Outcomes: Analyzing Binary Outcomes: An Introduction to Logistic Regression Robert Boudreau, PhD Co-Director of Methodology Core PITT-Multidisciplinary Clinical Research Center for Rheumatic and Musculoskeletal Diseases Core Director for Biostatistics Center for Aging and Population Health Center for Aging and Population Health Dept. of Epidemiology, GSPH 10/8/2010 Dept. of Epidemiology, GSPH 10/8/2010

Flow chart for group comparisons Measurements to be compared continuous Distribution approx normal or N ≥ 20? NoYes Non-parametrics T-tests discrete ( binary, nominal, ordinal with few values) Chi-square Fisher’s Exact

Flow chart for regression models (includes adjusted group comparisons) Outcome variable continuous or dichotomous? Dichotomous (binary)continuous Time-to-event available (or relevant)? NoYes Multiple logistic regression Cox proportional hazards regression Predictor variable categorical? NoYes (e.g. groups) Multiple linear regression ANCOVA (Multiple linear regression - using dummy variable(s) for categorical var(s)

Analysis From Last Didactic … In Health, Aging and Body Composition Knee-OA Substudy: In Health, Aging and Body Composition Knee-OA Substudy:  Examine Association between SxRxKOA (knee OA) and CRP adjusted for BMI. Motivation: Sowers M, Hochberg M et. al. C-reactive protein as a biomarker of emergent osteoarthritis. Osteoarthritis and Cartilage Volume 10, Issue 8, August 2002, Pages Conclusion: “CRP is highly associated with Knee OA; however, its high correlation with obesity limits its utility as an exclusive marker for knee OA”

Logistic Regression Outline for today Definition and interpretation of odds-ratio for binary outcome Essential equivalence of odds-ratio ↔ testing for group differences in rates (or percentages) when evaluated using 2 x 2 table, chi-square and p-values Logistic regression as “binary outcome” version of multiple linear regression: group (and covariate adjustment) effects are interpreted as odds-ratios affecting the binary outcome Detailed example: relating obesity to odds of knee OA - adjusted for race and gender

HABC: Obese x KneeOA Obese: BMI > 30 Chi-square P < Obese=1: Odds of kneeOA = p/(1-p)=0.2444/ = Obese=0: Odds of kneeOA = p/(1-p)=0.0911/ = Obesity odds-ratio for kneeOA OR = / =3.225

HABC: Obese x KneeOA proc logistic data=worst_knee_vs_noOA; model kneeOA(event="1")=obese; run; Note OR and C.I. Confidence Interval (C.I.) (2.56,4.04) doesn’t cover 1.0 => stat signif.

HABC: Obese x KneeOA Prob[kneeOA│obese=0]= exp(-2.3)/(1+exp(-2.3) = Prob[kneeOA│obese=0]= exp( )/(1+exp( ) =

HABC: Obese x KneeOA Obese: BMI > 30 Chi-square P < Prob[kneeOA│obese=0]= exp(-2.3)/(1+exp(-2.3) = Prob[kneeOA│obese=0]= exp( )/(1+exp( ) = General logistic regression form: Prob[kneeOA│obese] = exp(int+obese)/(1+exp(int+obese)

Gender x PAD

Gender x PAD (referent=female) proc logistic data=pad; model y1ppad(event=“1”)=male; run;

Gender x PAD (ref=male) proc logistic data=pad; model y1ppad(event=“1”)=female; run;

Gender x PAD (compare models: ref=female vs ref=male) (vs females) Male OR= (vs males) Female OR= = 1/1.891

CHD x KneeOA CHD  Knee OA association not statistically significant C.I.=(0.79,1.34)

Self-reported rheumatoid arthritis as binary outcome (or covariate) for analyses ? (NOT ?#!) Self-reported rheumatoid arthritis as binary outcome (or covariate) for analyses ? (NOT ?#!)

White Females: Obesity x KneeOA

White vs Black Females Obesity x KneeOA: Similar OR’s White Females Black Females

Black females have about two times higher rates of kneeOA than white women proc logistic data=worst_knee_vs_noOA; model kneeOA(event="1")= black ; where female; run;

Obesity odds-ratio is same for white and black women (interaction term is NS) proc logistic data=worst_knee_vs_noOA; model kneeOA(event="1")=obese black obese_x_black; where female; run;

Non-obese black women have OR=1.53 higher rates of knee OA, but obesity is associated with increased OR=3.61 for knee OA that applies within each race

Obesity explains some, but not all of the difference in rates of knee OA between black and white females (Note the “black race” OR attenuation from 2.08 to 1.53 after “adjusting” for obesity) model kneeOA= black obese

White Females: White Females: Continuous CRP Difference in average logCRP: 0.76 – 0.43 = 0.33 Knee OA P-value No (n=752)Yes (n=92) Mean (SD) Equal varsUnequal logCRP0.43 (0.83)0.76 (0.58)0.0002< logCRP SD’s were signif diff (p<0.0001) => Use Satterthwaite unequal variance test

All White Females in HABC (N=844) [includes SxRxKOA (n=93); also rest of parent study cohort] N=5 N=5 had CRP > 30 (max=63.2)

log CRP

White Females Continuous CRP as predictor of kneeOA Standardized var: mean-centered, divided by SD logCRP_perSD= (logCRP )/  Units of standardized logCRP is SD’s

White Females: Per SD higher logCRP, rates of knee OA increase by OR=1.5 proc logistic data=worst_knee_vs_noOA3; model kneeOA(event="1")=logCRP_perSD ; where female and white; run;

Thank you Questions, comments, suggestions or insights? Questions, comments, suggestions or insights? Remaining time: Open consultation … Remaining time: Open consultation …